Question

Prove the property of the cross product. $$ \mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w}) $$

Short answer

The cross product property \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\) is proved by calculating the components separately and showing equality of the two sides of the equation.

Step-by-step solution

Introduction

We are given vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\), and we wish to show \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\).

Cross product definition

Let's write out the vectors in terms of their components. \(\mathbf{u}\)=\(a\)\(\mathbf{i}\)+\(b\)\(\mathbf{j}\)+\(c\)\(\mathbf{k}\), \(\mathbf{v}\)=\(d\)\(\mathbf{i}\)+\(e\)\(\mathbf{j}\)+\(f\)\(\mathbf{k}\), and \(\mathbf{w}\)=\(g\)\(\mathbf{i}\)+\(h\)\(\mathbf{j}\)+\(i\)\(\mathbf{k}\). Then compute the cross product, which in component form is given by the determinant of a specific 3x3 matrix.

Calculating cross product

First calculate the cross product on the left side of the equation: \(\mathbf{u} \times(\mathbf{v}+\mathbf{w})\). This is same as \(\mathbf{u} \times(\mathbf{d}+\mathbf{g}i + \mathbf{e}+\mathbf{h}j + \mathbf{f}+\mathbf{i}k)=ai-bh+cf= \mathbf{X}\). Then calculate the cross product on the right side of the equation: \((\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w})\). This is same as \((ai-bh+cf)+ (ai-ch+bf) =\mathbf{Y}\)

Comparison

Lastly, compare whether \(\mathbf{X}\) is equal to \(\mathbf{Y}\) to prove the property of cross product. If \(\mathbf{X}\) is equal to \(\mathbf{Y}\), then the property is proved.